m

On the other hand, for the generalized gamma distribution (5.2) we have

a

x

h(x, f ) ¼ ap þ a : (5:8)

b

A direct comparison shows that (5.7) de¬nes a generalized gamma distribution

with À1 ¼ a and a ¼ Àap, an inverse generalized gamma distribution in our

terminology.

This shows that the (inverse) generalized gamma distribution can be derived from three

salient features of an income distribution, all of which can be veri¬ed from empirical data.

5.1.3 Moments and Other Basic Properties

Like the c.d.f. of the standard gamma distribution (5.1) (see below), the c.d.f. of the

generalized gamma distribution can be expressed in terms of Kummer™s con¬‚uent

hypergeometric function

X (c1 ) xn

1

n

1 F1 (c1 ; c2 ; x) ¼ , (5:9)

(c2 )n n!

n¼0

where (c)n ¼ c(c þ 1)(c þ 2) Á Á Á (c þ n À 1) is Pochhammer™s symbol, in the form

a !

a

eÀ(x=b) (x=b)ap x

F(x) ¼ 1 F1 1; p þ 1; , x ! 0: (5:10)

b

G( p þ 1)

Equivalently, it can be expressed in terms of an incomplete gamma function ratio

°

1 z pÀ1 Àt

F(x) ¼ t e dt, x ! 0, (5:11)

G( p) 0

Ðz

where z ¼ (x=b)a . Here g(n, z) ¼ 0 t nÀ1 eÀt dt is often called an incomplete gamma

function, although in the statistical literature this name is sometimes also used in

connection with (5.11).

The moments of the distribution (5.2) are given by

bk G( p þ k=a)

k

:

E(X ) ¼ (5:12)

G( p)

Hence, the ¬rst moment is

bG( p þ 1=a)

E(X ) ¼ (5:13)

G( p)

152 GAMMA-TYPE SIZE DISTRIBUTIONS

and the variance equals

& '

G( p þ 2=a)G( p) À [G( p þ 1=a)]2

var(X ) ¼ b2 : (5:14)

[G( p)]2

The expressions for the skewness and kurtosis coef¬cients are rather lengthy and

therefore not given here. It is however interesting that there is a value a ¼ a( p) for

p¬¬¬¬¬ p¬¬¬¬¬ p¬¬¬¬¬

which the shape factor b1 ¼ 0. For a , a( p), b1 , 0; for a . a( p), b1 . 0.

This property of the generalized gamma distribution is inherited by the Weibull

distribution discussed below. Figures 5.2 and 5.3 depict some generalized gamma

densities, including left-skewed examples.

As noted in the preceding chapter, the best known distribution that is not

determined by the sequence of its moments (despite all the moments being ¬nite) is

the lognormal distribution. Pakes and Khattree (1992) showed that the generalized

gamma distribution provides a further example of a distribution possessing this

somewhat pathological and unexpected property. Speci¬cally, the distribution is

determined by the moments only if a . 1 , whereas for a 1, any distribution with

2 2

p.d.f.

f (x){1 À e sin(2pap þ xa tan 2pa)} (5:15)

e , 1.

has the same moments for À1

Figure 2 Generalized gamma densities: a ¼ 8, b ¼ 1; and p ¼ 0:25, 0:5, 1, 2, 4 (from left to right).

153

5.1 GENERALIZED GAMMA DISTRIBUTION

Figure 3 Generalized gamma densities: p ¼ 4, b ¼ 1; and a ¼ 1, 1:5, 2, 2:5, 3 (from right to left).

The mode of the generalized gamma distribution occurs at

1=a

1

¼b pÀ , for ap . 1:

xmode (5:16)

a

Otherwise, the distribution is zeromodal with a pole at the origin if ap , 1.

The generalized gamma distribution allows for a wide array of shapes of the

hazard rate. The situation is best analyzed utilizing general results due to Glaser

(1980). He considered the reciprocal hazard rate

1 1 À F(x)

g(x) ¼ ¼

r(x) f (x)

whose derivative is

g0 (x) ¼ g(x)q(x) À 1,

where

f 0 (x)

q(x) ¼ À :

f (x)

The shape of r(x) now depends on the behavior of q 0 . It is not dif¬cult to see that

q 0 (x) . 0 (q 0 (x) , 0), for all x . 0, implies an increasing (decreasing) hazard rate.

154 GAMMA-TYPE SIZE DISTRIBUTIONS

If q 0 changes signs, with q 0 (x0 ) ¼ 0 for some x0 . 0, and q 0 (x) , 0 for x , x0 and

q 0 (x) . 0 for x . x0, we have an increasing hazard rate if limx!0 f (x) ¼ 0 and a

S

T-shaped hazard rate if limx!0 f (x) ¼ 1. Similarly, we obtain decreasing and

-shaped hazard rates if the inequalities in the preceding conditions are reversed.

For the generalized gamma distribution the function q is

1 À ap axaÀ1

q(x) ¼ þ a:

b

x

Hence,

xa [a(a À 1)] þ (ap À 1)ba

0

q (x) ¼ :

x 2 ba